The letters A,B,C,D,E,F, and G are arranged in random order.

Question

3. The letters A,B,C,D,E,F, and G are arranged in random order.

(a) What is the probability that A and E are next to each other?

(b) What is the probability that A, B, and C appearing in a row in some

order in the string of letters?

Answer 1

Hi Julie,

 

There are 7! ways that the letters A through G can be arranged in random order.

 

7! = 7.6.5.4.3.2.1

 

If we keep A and E next to each other then A and E must always be in adjacent positions

 

so A could be in postion 1 and E could be in postion 2

 

or A could be in position 2 and E could be in position 3

 

This can happen six different ways but we could also switch A and E so we must mutliply by 2.

 

So there are 12 ways that A and E can be placed next to each other if we have a total of 7 positions.

 

now the other five numbers can be positioned randomly so that would be 5!.

 

the total number of ways to have A and E next to each other would be 12 * 5!

 

and the probability of this happening is that number divided by 7!

 

so the answer is (12*5!)/7!

 

Part B can be solved in a similar fashion give it a try.

 

Hope this helps